# The Monad.Reader/Issue5/Number Param Types

### From HaskellWiki

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dependent-type-like system that is just a Haskell library. The basics |
dependent-type-like system that is just a Haskell library. The basics |
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of the number-parameterized types are written in Haskell98. |
of the number-parameterized types are written in Haskell98. |
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+ | |||

=== Keywords: === |
=== Keywords: === |
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Haskell, number-parameterized types, type arithmetic, decimal types, type-directed programming. |
Haskell, number-parameterized types, type arithmetic, decimal types, type-directed programming. |
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+ | |||

= Contents = |
= Contents = |
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be checked across module boundaries. |
be checked across module boundaries. |
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− | In this paper, we develop realizations of number-parameterized |
+ | In this paper, we develop realizations of number-parameterized types in Haskell that indeed have all the above advantages. The numeric parameter is specified in ''decimal'' rather than in binary, which makes types smaller and far easier to read. Type error messages also become more comprehensible. The programmer may write or the compiler can infer equality constraints (e.g., two argument |

− | types in Haskell that indeed have all the above advantages. The |
+ | vectors of a function must be of the same size), arithmetic constraints (e.g., one vector must be larger by some amount), and inequality constraints (e.g., the size of the argument vector must be at least one). The violations of the constraints are detected at compile time. We can remove run-time tag checks in functions like <hask>vhead</hask>, which are statically assured to receive a non-empty vector. |

− | numeric parameter is specified in ''decimal'' rather than in |
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− | binary, which makes types smaller and far easier to read. Type error |
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− | messages also become more comprehensible. The programmer may write or |
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− | the compiler can infer equality constraints (e.g., two argument |
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− | vectors of a function must be of the same size), arithmetic |
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− | constraints (e.g., one vector must be larger by some amount), and |
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− | inequality constraints (e.g., the size of the argument vector must be |
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− | at least one). The violations of the constraints are detected at |
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− | compile time. We can remove run-time tag checks in functions like |
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− | <hask>vhead</hask>, which are statically assured to receive a non-empty |
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− | vector. |
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Although we come close to the dependent-type programming, we do |
Although we come close to the dependent-type programming, we do |
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types. |
types. |
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− | Our running example is arrays parameterized over their size. The |
+ | Our running example is arrays parameterized over their size. The parameter of the vector type is therefore a non-negative integer number. For simplicity, all the vectors in the paper are indexed from zero. In addition to vector constructors and element accessors, we define a <hask>zipWith</hask>-like operation to map two vectors onto the third, element by element. An attempt to map vectors of different sizes should be reported as a type error. The typechecker will also guarantee that there is no attempt to allocate a vector of a negative size. In Section [[#Computations with decimal types| sec:arithmetic]] we introduce operations <hask>vhead</hask>, <hask>vtail</hask> and <hask>vappend</hask> on number-parameterized vectors. The types of these operations exhibit arithmetic and inequality constraints. |

− | parameter of the vector type is therefore a non-negative integer |
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− | number. For simplicity, all the vectors in the paper are indexed from |
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− | zero. In addition to vector constructors and element accessors, we |
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− | define a <hask>zipWith</hask>-like operation to map two vectors onto |
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− | the third, element by element. An attempt to map vectors of different |
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− | sizes should be reported as a type error. The typechecker will also |
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− | guarantee that there is no attempt to allocate a vector of a negative |
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− | size. In Section [[#Computations with decimal types|sec:arithmetic]] we introduce operations <hask>vhead</hask>, <hask>vtail</hask> and <hask>vappend</hask> on number-parameterized vectors. |
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− | The types of these operations exhibit arithmetic and inequality |
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− | constraints. |
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The present paper describes several gradually more sophisticated |
The present paper describes several gradually more sophisticated |
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+ | == Encoding the number parameter in data constructors == |
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− | == Encoding the number parameter in data constructors == |
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The first approach to vectors parameterized by their size encodes |
The first approach to vectors parameterized by their size encodes |
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the size as a series of data constructors. This approach has been used |
the size as a series of data constructors. This approach has been used |
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following sections. |
following sections. |
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− | Our representation of vectors of a statically checked size is |
+ | Our representation of vectors of a statically checked size is reminiscent of the familiar representation of lists: |

− | reminiscent of the familiar representation of lists: |
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<haskell> |
<haskell> |
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data List a = Nil | Cons a (List a) |
data List a = Nil | Cons a (List a) |
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</haskell> |
</haskell> |
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− | <hask>List a</hask> is a recursive datatype. Lists of different sizes |
+ | <hask>List a</hask> is a recursive datatype. Lists of different sizes have the same recursive type. To make the types different (so that we can represent the size, too) we break the explicit recursion in the datatype declaration. We introduce two data constructors: |

− | have the same recursive type. To make the types different (so that |
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− | we can represent the size, too) we break the explicit recursion in the |
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− | datatype declaration. We introduce two data constructors: |
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<haskell> |
<haskell> |
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module UnaryDS where |
module UnaryDS where |
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data Vecp tail a = a :+: (tail a) deriving Show |
data Vecp tail a = a :+: (tail a) deriving Show |
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</haskell> |
</haskell> |
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− | The constructor <hask>VZero</hask> represents a vector of a zero |
+ | The constructor <hask>VZero</hask> represents a vector of a zero size. A value of the type <hask>Vecp tail a</hask> is a non-empty vector formed from an element of the type <hask>a</hask> and (a smaller vector) of the type <hask>tail a</hask>. We place our vectors into the class <hask>Show</hask> for expository purposes. Thus vectors holding one element have the type <hask>Vecp VZero a</hask>, vectors with two elements have the type <hask>Vecp (Vecp VZero) a</hask>, with three elements <hask>Vecp (Vecp (Vecp VZero)) a</hask>, etc. We should stress the separation of the shape type of a vector, <hask>Vecp (Vecp VZero)</hask> in the last example, from the type of vector elements. The shape |

− | size. A value of the type <hask>Vecp tail a</hask> is a non-empty vector |
+ | type of a vector clearly encodes vector’s size, as repeated applications of a type constructor <hask>Vecp</hask> to the type constructor <hask>VZero</hask>, i.e., as a Peano numeral. We have indeed designed a number-parameterized ''type''. |

− | formed from an element of the type <hask>a</hask> and (a smaller vector) |
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− | of the type <hask>tail a</hask>. We place our vectors into the class |
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− | <hask>Show</hask> for expository purposes. Thus vectors holding one |
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− | element have the type <hask>Vecp VZero a</hask>, vectors with two |
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− | elements have the type <hask>Vecp (Vecp VZero) a</hask>, with three elements |
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− | <hask>Vecp (Vecp (Vecp VZero)) a</hask>, etc. We should stress the |
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− | separation of the shape type of a vector, <hask>Vecp (Vecp VZero)</hask> in the last example, from the type of vector elements. The shape |
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− | type of a vector clearly encodes vector’s size, as repeated |
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− | applications of a type constructor <hask>Vecp</hask> to the type |
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− | constructor <hask>VZero</hask>, i.e., as a Peano numeral. We have indeed |
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− | designed a number-parameterized ''type''. |
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− | To generically manipulate the family of differently-sized vectors, |
+ | To generically manipulate the family of differently-sized vectors, we define a class of polymorphic functions: |

− | we define a class of polymorphic functions: |
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<haskell> |
<haskell> |
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class Vec t where |
class Vec t where |
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vzipWith:: (a->b->c) -> t a -> t b -> t c |
vzipWith:: (a->b->c) -> t a -> t b -> t c |
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</haskell> |
</haskell> |
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− | The method <hask>vlength</hask> gives us the size of a vector; the |
+ | The method <hask>vlength</hask> gives us the size of a vector; the method <hask>vat</hask> lets us retrieve a specific element, and the method <hask>vzipWith</hask> produces a vector by an element-by-element combination of two other vectors. We can use <hask>vzipWith</hask> to add two vectors elementwise. We must emphasize the type of <hask>vzipWith</hask>: the two argument vectors may hold elements of different types, but the vectors must have the same shape, i.e., size. |

− | method <hask>vat</hask> lets us retrieve a specific element, and the method |
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− | <hask>vzipWith</hask> produces a vector by an element-by-element |
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− | combination of two other vectors. We can use <hask>vzipWith</hask> to |
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− | add two vectors elementwise. We must emphasize the type of <hask>vzipWith</hask>: the two argument vectors may hold elements of different |
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− | types, but the vectors must have the same shape, i.e., size. |
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− | The implementation of the class <hask>Vec</hask> has only two |
+ | The implementation of the class <hask>Vec</hask> has only two instances: |

− | instances: |
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<haskell> |
<haskell> |
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instance Vec VZero where |
instance Vec VZero where |
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(f a b) :+: (vzipWith f ta tb) |
(f a b) :+: (vzipWith f ta tb) |
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</haskell> |
</haskell> |
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− | The second instance makes it clear that a value of a type <hask>Vecp tail a</hask> is a vector <hask>Vec</hask> if and only if |
+ | The second instance makes it clear that a value of a type <hask>Vecp tail a</hask> is a vector <hask>Vec</hask> if and only if <hask>tail a</hask> is a vector <hask>Vec</hask>. Our vectors, instances of the class <hask>Vec</hask>, are recursively defined too. Unlike lists, our vectors reveal their sizes in their types. |

− | <hask>tail a</hask> is a vector <hask>Vec</hask>. Our vectors, |
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− | instances of the class <hask>Vec</hask>, are recursively defined too. Unlike |
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− | lists, our vectors reveal their sizes in their types. |
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− | That was the complete implementation of the number-parameterized |
+ | That was the complete implementation of the number-parameterized vectors. We can now define a few sample vectors: |

− | vectors. We can now define a few sample vectors: |
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<haskell> |
<haskell> |
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v3c = 'a' :+: 'b' :+: 'c' :+: VZero |
v3c = 'a' :+: 'b' :+: 'c' :+: VZero |
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test2 = [vat v3c 0, vat v3c 1, vat v3c 2] |
test2 = [vat v3c 0, vat v3c 1, vat v3c 2] |
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</haskell> |
</haskell> |
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− | We can load the code into a Haskell system and run the |
+ | We can load the code into a Haskell system and run the tests. Incidentally, we can ask the Haskell system to tell us the inferred type of a sample vector: |

− | tests. Incidentally, we can ask the Haskell system to tell us the |
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− | inferred type of a sample vector: |
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<haskell> |
<haskell> |
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*UnaryDS> :t v3c |
*UnaryDS> :t v3c |
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Vecp (Vecp (Vecp VZero)) Char |
Vecp (Vecp (Vecp VZero)) Char |
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</haskell> |
</haskell> |
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− | The inferred type indeed encodes the size of the vector as a |
+ | The inferred type indeed encodes the size of the vector as a Peano numeral. We can try more complex tests, of element-wise operations on two vectors: |

− | Peano numeral. We can try more complex tests, of element-wise |
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− | operations on two vectors: |
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<haskell> |
<haskell> |
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test3 = vzipWith (\c i -> (toEnum $ fromEnum c + fromIntegral i)::Char) |
test3 = vzipWith (\c i -> (toEnum $ fromEnum c + fromIntegral i)::Char) |
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'b' :+: ('d' :+: ('f' :+: VZero)) |
'b' :+: ('d' :+: ('f' :+: VZero)) |
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</haskell> |
</haskell> |
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− | In particular, <hask>test3</hask> demonstrates an operation on two |
+ | In particular, <hask>test3</hask> demonstrates an operation on two vectors of the same shape but of different element types. |

− | vectors of the same shape but of different element types. |
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− | An attempt to add, by mistake, two vectors of different sizes is |
+ | An attempt to add, by mistake, two vectors of different sizes is revealing: |

− | revealing: |
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<haskell> |
<haskell> |
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test5 = vzipWith (+) v3i v4i |
test5 = vzipWith (+) v3i v4i |
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In the definition of `test5': vzipWith (+) v3i v4i |
In the definition of `test5': vzipWith (+) v3i v4i |
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</haskell> |
</haskell> |
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− | We get a type error, with a clear error message (the quoted message, |
+ | We get a type error, with a clear error message (the quoted message, here and elsewhere in the paper, is by GHCi. The Hugs error message is essentially the same). The typechecker, at the compile time, has detected that the sizes of the vectors to add elementwise do not match. To be more precise, the sizes are off by one. |

− | here and elsewhere in the paper, is by GHCi. The Hugs error message |
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− | is essentially the same). The typechecker, at the compile time, has |
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− | detected that the sizes of the vectors to add elementwise do not |
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− | match. To be more precise, the sizes are off by one. |
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− | For vectors described in this section, the element access |
+ | For vectors described in this section, the element access operation, <hask>vat</hask>, takes <tt>O(n)</tt> time where <tt>n</tt> is the size of the vector. [[#Okasaki99|Chris Okasaki]] has designed more sophisticated number-parameterized vectors with element access time <tt>O(log n)</tt>. Although this is an improvement, the overhead of accessing an element adds up for many operations. Furthermore, the overhead of data constructors, <hask>:+:</hask> in our example, becomes noticeable for longer vectors. When we encode the size of a vector as a sequence of data constructors, the latter overhead cannot be eliminated. |

− | operation, <hask>vat</hask>, takes <tt>O(n)</tt> time where |
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− | <tt>n</tt> is the size of the vector. [[#Okasaki99|Chris Okasaki]] has designed more sophisticated number-parameterized |
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− | vectors with element access time <tt>O(log n)</tt>. Although this |
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− | is an improvement, the overhead of accessing an element adds up for |
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− | many operations. Furthermore, the overhead of data constructors, |
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− | <hask>:+:</hask> in our example, becomes noticeable for longer |
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− | vectors. When we encode the size of a vector as a sequence of data |
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− | constructors, the latter overhead cannot be eliminated. |
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− | |||

− | Although we have achieved the separation of the shape type of a |
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− | vector from the type of its elements, we did so at the expense of a |
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− | sequence of data constructors, <hask>:+:</hask>, at the term |
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− | level. These constructors add time and space overheads, which |
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− | increase with the vector size. In the following sections we |
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− | show more efficient representations for number-parameterized |
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− | vectors. The structure of their type will still tell us the size of |
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− | the vector; however there will be no corresponding term structure, |
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− | and, therefore, no space overhead of storing it nor run-time overhead |
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− | of traversing it. |
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+ | Although we have achieved the separation of the shape type of a vector from the type of its elements, we did so at the expense of a sequence of data constructors, <hask>:+:</hask>, at the term level. These constructors add time and space overheads, which increase with the vector size. In the following sections we show more efficient representations for number-parameterized vectors. The structure of their type will still tell us the size of the vector; however there will be no corresponding term structure, and, therefore, no space overhead of storing it nor run-time overhead of traversing it. |
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== Encoding the number parameter in type constructors, in unary == |
== Encoding the number parameter in type constructors, in unary == |
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− | To improve the efficiency of number-parameterized vectors, we |
+ | |

− | choose a better run-time representation: Haskell arrays. The code in |
+ | To improve the efficiency of number-parameterized vectors, we choose a better run-time representation: Haskell arrays. The code in the present section is in Haskell98. |

− | the present section is in Haskell98. |
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<haskell> |
<haskell> |
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module UnaryT (..elided..) where |
module UnaryT (..elided..) where |
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import Data.Array |
import Data.Array |
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</haskell> |
</haskell> |
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− | First, we need a type structure (an infinite family of types) to |
+ | First, we need a type structure (an infinite family of types) to encode non-negative numbers. In the present section, we will use an unary encoding in the form of Peano numerals. The unary type encoding of integers belongs to programming folklore. It is also described in [[#Blume01|Blume01]] in the context of a foreign-function interface library of SML. |

− | encode non-negative numbers. In the present section, we will use an |
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− | unary encoding in the form of Peano numerals. The unary type encoding of |
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− | integers belongs to programming folklore. It is also described in |
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− | [[#Blume01|Blume01]] in the context of a foreign-function interface |
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− | library of SML. |
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<haskell> |
<haskell> |
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data Zero = Zero |
data Zero = Zero |
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data Succ a = Succ a |
data Succ a = Succ a |
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</haskell> |
</haskell> |
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− | That is, the term <hask>Zero</hask> of the type <hask>Zero</hask> |
+ | That is, the term <hask>Zero</hask> of the type <hask>Zero</hask> represents the number 0. The term <hask>(Succ (Succ Zero))</hask> of the type <hask>(Succ (Succ Zero))</hask> encodes the number two. We call these numerals Peano numerals because the number <tt>n</tt> is represented as a repeated application of <tt>n</tt> type (data) constructors <hask>Succ</hask> to the type (term) <hask>Zero</hask>. We observe a one-to-one correspondence between the types of our numerals and the terms. In fact, a numeral term looks precisely the same as its type. This property is crucial as we shall see on many occasions below. It lets us “lift” number computations to the type level. The property also makes error messages lucid <ref>We could have declared <hask>Succ</hask> as <hask>newtype Succ a = Succ a</hask> so that <hask>Succ</hask> is just a tag and all non-bottom Peano numerals share the same run-time representation. As we shall see however, we hardly ever use the values of our numerals.</ref>. |

− | represents the number 0. The term <hask>(Succ (Succ Zero))</hask> of the type |
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− | <hask>(Succ (Succ Zero))</hask> encodes the number two. We call these |
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− | numerals Peano numerals because the number <tt>n</tt> is |
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− | represented as a repeated application of <tt>n</tt> type (data) |
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− | constructors <hask>Succ</hask> to the type (term) <hask>Zero</hask>. We observe a one-to-one correspondence between the types of our |
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− | numerals and the terms. In fact, a numeral term looks precisely the |
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− | same as its type. This property is crucial as we shall see on many |
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− | occasions below. It lets us “lift” number computations to the type |
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− | level. The property also makes error messages lucid |
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− | <ref>We could have declared <hask>Succ</hask> as |
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− | <hask>newtype Succ a = Succ a</hask> so that <hask>Succ</hask> is just a |
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− | tag and all non-bottom Peano numerals share the same run-time |
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− | representation. As we shall see however, we hardly ever use the values of |
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− | our numerals.</ref>. |
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− | We place our Peano numerals into a class <hask>Card</hask>, which |
+ | We place our Peano numerals into a class <hask>Card</hask>, which has a method <hask>c2num</hask> to convert a numeral into the corresponding number. |

− | has a method <hask>c2num</hask> to convert a numeral into the |
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− | corresponding number. |
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<haskell> |
<haskell> |
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class Card c where |
class Card c where |
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Line 228: | Line 231: | ||

c2num x = 1 + c2num (cpred x) |
c2num x = 1 + c2num (cpred x) |
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</haskell> |
</haskell> |
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− | The function <hask>cpred</hask> determines the predecessor for a |
+ | The function <hask>cpred</hask> determines the predecessor for a positive Peano numeral. The definition for that function may seem puzzling: it is undefined. We observe that the callers do not need the value returned by that function: they merely need the type of that value. Indeed, let us examine the definitions of the method <hask>c2num</hask> in the above two instances. In the instance <hask>Card Zero</hask>, we are certain that the argument of <hask>c2num</hask> has the type <hask>Zero</hask>. That type, in our encoding, represents the number zero, which we return. There can be only one non-bottom value of the type <hask>Zero</hask>: therefore, once we know the type, we do not need to examine the value. Likewise, in the instance <hask>Card (Succ c)</hask>, we know that the type of the argument of <hask>c2num</hask> is <hask>(Succ c)</hask>, where <hask>c</hask> is itself a <hask>Card</hask> numeral. If we could convert a value of the type <hask>c</hask> to a number, we can convert the value of the type <hask>(Succ c)</hask> as well. By induction we determine that <hask>c2num</hask> never examines the value of its argument. Indeed, not only <hask>c2num (Succ (Succ Zero))</hask> evaluates to 2, but so does <hask>c2num (undefined::(Succ (Succ Zero)))</hask>. |

− | positive Peano numeral. The definition for that function may seem |
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− | puzzling: it is undefined. We observe that the callers do not need the value |
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− | returned by that function: they merely need the type of that |
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− | value. Indeed, let us examine the definitions of the method <hask>c2num</hask> in the above two instances. In the instance <hask>Card Zero</hask>, we are certain that the argument of <hask>c2num</hask> has |
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− | the type <hask>Zero</hask>. That type, in our encoding, represents the |
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− | number zero, which we return. There can be only one non-bottom value |
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− | of the type <hask>Zero</hask>: therefore, once we know the type, we do |
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− | not need to examine the value. Likewise, in the instance |
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− | <hask>Card (Succ c)</hask>, we know that the type of the argument of <hask>c2num</hask> is <hask>(Succ c)</hask>, where <hask>c</hask> is itself a |
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− | <hask>Card</hask> numeral. If we could convert a value of the type |
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− | <hask>c</hask> to a number, we can convert the value of the type <hask>(Succ c)</hask> as well. By induction we determine that <hask>c2num</hask> never examines the value of its argument. Indeed, not only <hask>c2num (Succ (Succ Zero))</hask> evaluates to 2, but so does |
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− | <hask>c2num (undefined::(Succ (Succ Zero)))</hask>. |
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− | The same correspondence between the types and the terms suggests |
+ | The same correspondence between the types and the terms suggests that the numeral type alone is enough to describe the size of a vector. We do not need to store the value of the numeral. The shape type of our vectors could be ''phantom'' (as in [[#Blume01|Blume01]]). |

− | that the numeral type alone is enough to describe the size of a |
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− | vector. We do not need to store the value of the numeral. The shape |
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− | type of our vectors could be ''phantom'' (as in [[#Blume01|Blume01]]). |
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<haskell> |
<haskell> |
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newtype Vec size a = Vec (Array Int a) deriving Show |
newtype Vec size a = Vec (Array Int a) deriving Show |
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</haskell> |
</haskell> |
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− | That is, the type variable <hask>size</hask> does not occur on the |
+ | That is, the type variable <hask>size</hask> does not occur on the right-hand size of the <hask>Vec</hask> declaration. More importantly, at run-time our <hask>Vec</hask> is indistinguishable from an <hask>Array</hask>, thus incurring no additional overhead and providing constant-time element access. As we mentioned earlier, for simplicity, all the vectors in the paper are indexed from zero. The data constructor <hask>Vec</hask> is not exported from the module, so one has to use the following functions to construct vectors. |

− | right-hand size of the <hask>Vec</hask> declaration. More importantly, |
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− | at run-time our <hask>Vec</hask> is indistinguishable from an <hask>Array</hask>, thus incurring no additional overhead and providing |
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− | constant-time element access. As we mentioned earlier, for simplicity, |
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− | all the vectors in the paper are indexed from zero. The data |
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− | constructor <hask>Vec</hask> is not exported from the module, so one |
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− | has to use the following functions to construct vectors. |
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<haskell> |
<haskell> |
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listVec':: (Card size) => size -> [a] -> Vec size a |
listVec':: (Card size) => size -> [a] -> Vec size a |
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Line 247: | Line 250: | ||

vec size elem = listVec' size $ repeat elem |
vec size elem = listVec' size $ repeat elem |
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</haskell> |
</haskell> |
||

− | The private function <hask>listVec'</hask> constructs the vector |
+ | The private function <hask>listVec'</hask> constructs the vector of the requested size initialized with the given values. The function makes no check that the length of the list of the initial values <hask>elems</hask> is equal to the length of the vector. We use this non-exported function internally, when we have proven that <hask>elems</hask> has the right length, or when truncating such a list is appropriate. The exported function <hask>listVec</hask> is a safe version of <hask>listVec'</hask>. The former assures that the constructed vector is consistently initialized. The function <hask>vec</hask> initializes all elements to the same value. For example, the following expression creates a boolean vector of two elements with the initial values <hask>True</hask> and <hask>False</hask>. |

− | of the requested size initialized with the given values. The function |
||

− | makes no check that the length of the list of the initial values |
||

− | <hask>elems</hask> is equal to the length of the vector. We use this |
||

− | non-exported function internally, when we have proven that <hask>elems</hask> has the right length, or when truncating such a list is |
||

− | appropriate. The exported function <hask>listVec</hask> is a safe |
||

− | version of <hask>listVec'</hask>. The former assures that the |
||

− | constructed vector is consistently initialized. The function <hask>vec</hask> initializes all elements to the same value. For example, the |
||

− | following expression creates a boolean vector of two elements with the |
||

− | initial values <hask>True</hask> and <hask>False</hask>. |
||

<haskell> |
<haskell> |
||

*UnaryT> listVec (Succ (Succ Zero)) [True,False] |
*UnaryT> listVec (Succ (Succ Zero)) [True,False] |
||

Vec (array (0,1) [(0,True),(1,False)]) |
Vec (array (0,1) [(0,True),(1,False)]) |
||

</haskell> |
</haskell> |
||

− | A Haskell interpreter created the requested value, and printed it |
+ | A Haskell interpreter created the requested value, and printed it out. We can confirm that the inferred type of the vector encodes its size: |

− | out. We can confirm that the inferred type of the vector encodes its |
||

− | size: |
||

<haskell> |
<haskell> |
||

*UnaryT> :type listVec (Succ (Succ Zero)) [True,False] |
*UnaryT> :type listVec (Succ (Succ Zero)) [True,False] |
||

Vec (Succ (Succ Zero)) Bool |
Vec (Succ (Succ Zero)) Bool |
||

</haskell> |
</haskell> |
||

− | We can now introduce functions to operate on our vectors. The |
+ | We can now introduce functions to operate on our vectors. The functions are similar to those in the previous section. As before, they are polymorphic in the shape of vectors (i.e., their sizes). This polymorphism is expressed differently however. In the present section we use just the parametric polymorphism rather than typeclasses. |

− | functions are similar to those in the previous section. As before, |
||

− | they are polymorphic in the shape of vectors (i.e., their sizes). This |
||

− | polymorphism is expressed differently however. In the present section |
||

− | we use just the parametric polymorphism rather than typeclasses. |
||

<haskell> |
<haskell> |
||

vlength_t:: Vec size a -> size |
vlength_t:: Vec size a -> size |
||

Line 275: | Line 278: | ||

listVec' (vlength_t va) $ zipWith f (velems va) (velems vb) |
listVec' (vlength_t va) $ zipWith f (velems va) (velems vb) |
||

</haskell> |
</haskell> |
||

− | The functions <hask>vlength_t</hask> and <hask>vlength</hask> tell |
+ | The functions <hask>vlength_t</hask> and <hask>vlength</hask> tell the size of their argument vector. The function <hask>vat</hask> returns the element of a vector at a given zero-based index. The function <hask>velems</hask>, which gives the list of vector’s elements, is the left inverse of <hask>listVec</hask>. The function <hask>vzipWith</hask> elementwise combines two vectors into the third one by applying a user-specified function <hask>f</hask> to the |

− | the size of their argument vector. The function <hask>vat</hask> |
+ | corresponding elements of the argument vectors. The polymorphic types of these functions indicate that the functions generically operate on number-parameterized vectors of any <hask>size</hask>. Furthermore, the type of <hask>vzipWith</hask> expresses the constraint that the two argument vectors must have the same size. The result will be a vector of the same size as that of the argument vectors. We rely on the fact that the function <hask>zipWith</hask>, when applied to two lists of the same size, gives the list of that size. This justifies our use of <hask>listVec'</hask>. |

− | returns the element of a vector at a given zero-based index. The function |
||

− | <hask>velems</hask>, which gives the list of vector’s elements, is the |
||

− | left inverse of <hask>listVec</hask>. The function |
||

− | <hask>vzipWith</hask> elementwise combines two vectors into the third |
||

− | one by applying a user-specified function <hask>f</hask> to the |
||

− | corresponding elements of the argument vectors. The polymorphic types |
||

− | of these functions indicate that the functions generically operate on |
||

− | number-parameterized vectors of any <hask>size</hask>. Furthermore, |
||

− | the type of <hask>vzipWith</hask> expresses the constraint that the |
||

− | two argument vectors must have the same size. The result will be a |
||

− | vector of the same size as that of the argument vectors. We rely on |
||

− | the fact that the function <hask>zipWith</hask>, when applied to two |
||

− | lists of the same size, gives the list of that size. This justifies our |
||

− | use of <hask>listVec'</hask>. |
||

− | We have introduced two functions that yield the size of their |
+ | We have introduced two functions that yield the size of their argument vector. One is the function <hask>vlength_t</hask>: it returns a value whose type represents the size of the vector. We are interested only in the type of the return value — which we extract statically from the type of the argument vector. The function <hask>vlength_t</hask> is a ''compile-time'' function. Therefore, it is no surprise that its body is <hask>undefined</hask>. The type of the function ''is'' its true definition. The function <hask>vlength</hask> in contrast retrieves vector’s size from the run-time representation as an array. If we export <hask>listVec</hask> from the module <hask>UnaryT</hask> but do not export the constructor <hask>Vec</hask>, we can guarantee that <hask>c2num . vlength_t</hask> is equivalent to <hask>vlength</hask>: our number-parameterized vector type is sound. |

− | argument vector. One is the function <hask>vlength_t</hask>: it |
||

− | returns a value whose type represents the size of the vector. We are |
||

− | interested only in the type of the return value — which we extract |
||

− | statically from the type of the argument vector. The function <hask>vlength_t</hask> is a ''compile-time'' function. Therefore, it is |
||

− | no surprise that its body is <hask>undefined</hask>. The type of the |
||

− | function ''is'' its true definition. The function <hask>vlength</hask> in contrast retrieves vector’s size from the run-time |
||

− | representation as an array. If we export <hask>listVec</hask> from the |
||

− | module <hask>UnaryT</hask> but do not export the constructor <hask>Vec</hask>, we can guarantee that <hask>c2num . vlength_t</hask> is |
||

− | equivalent to <hask>vlength</hask>: our number-parameterized vector |
||

− | type is sound. |
||

− | From the practical point of view, passing terms such as |
+ | From the practical point of view, passing terms such as <hask>(Succ (Succ Zero))</hask> to the functions <hask>vec</hask> or <hask>listVec</hask> to construct vectors is inconvenient. The previous section showed a better approach. We can implement it here too: we let the user enumerate the values, which we accumulate into a list, counting them at the same time: |

− | <hask>(Succ (Succ Zero))</hask> to the functions <hask>vec</hask> or <hask>listVec</hask> to construct vectors is inconvenient. The previous |
||

− | section showed a better approach. We can implement it here too: we let |
||

− | the user enumerate the values, which we accumulate into a list, |
||

− | counting them at the same time: |
||

<haskell> |
<haskell> |
||

infixl 3 &+ |
infixl 3 &+ |
||

Line 291: | Line 294: | ||

vc (VC size lst) = listVec' size (reverse lst) |
vc (VC size lst) = listVec' size (reverse lst) |
||

</haskell> |
</haskell> |
||

− | The counting operation is effectively performed by a typechecker |
+ | The counting operation is effectively performed by a typechecker at compile time. Finally, the function <hask>vc</hask> will allocate and initialize the vector of the right size — and of the right type. Here are a few sample vectors and operations on them: |

− | at compile time. Finally, the function <hask>vc</hask> will allocate |
||

− | and initialize the vector of the right size — and of the right |
||

− | type. Here are a few sample vectors and operations on them: |
||

<haskell> |
<haskell> |
||

v3c = vc $ vs &+ 'a' &+ 'b' &+ 'c' |
v3c = vc $ vs &+ 'a' &+ 'b' &+ 'c' |
||

Line 310: | Line 313: | ||

Vec (Succ (Succ (Succ Zero))) Char |
Vec (Succ (Succ (Succ Zero))) Char |
||

</haskell> |
</haskell> |
||

− | The type of the result bears the clear indication of the size of |
+ | The type of the result bears the clear indication of the size of the vector. If we attempt to perform an element-wise operation on vectors of different sizes, for example: |

− | the vector. If we attempt to perform an element-wise operation on |
||

− | vectors of different sizes, for example: |
||

<haskell> |
<haskell> |
||

test5 = vzipWith (+) v3i v4i |
test5 = vzipWith (+) v3i v4i |
||

Line 319: | Line 322: | ||

In the definition of `test5': vzipWith (+) v3i v4i |
In the definition of `test5': vzipWith (+) v3i v4i |
||

</haskell> |
</haskell> |
||

− | we get a message from the typechecker that the sizes are off |
+ | we get a message from the typechecker that the sizes are off by one. |

− | by one. |
||

− | |||

== Fixed-precision decimal types == |
== Fixed-precision decimal types == |
||

− | Peano numerals adequately represent the size of a vector in vector’s |
||

− | type. However, they make the notation quite verbose. We want to offer |
||

− | a programmer a familiar, decimal notation for the terms and the types |
||

− | representing non-negative numerals. This turns out possible even in |
||

− | Haskell98. In this section, we describe a fixed-precision notation, |
||

− | assuming that a programmer will never need a vector with more than 999 |
||

− | elements. The limit is not hard and can be readily extended. The next |
||

− | section will eliminate the limit altogether. |
||

− | We again will be using Haskell arrays as the run-time |
+ | Peano numerals adequately represent the size of a vector in vector’s type. However, they make the notation quite verbose. We want to offer a programmer a familiar, decimal notation for the terms and the types representing non-negative numerals. This turns out possible even in Haskell98. In this section, we describe a fixed-precision notation, assuming that a programmer will never need a vector with more than 999 elements. The limit is not hard and can be readily extended. The next section will eliminate the limit altogether. |

− | representation for our vectors. In fact, the implementation of |
+ | |

− | vectors is the same as that in the previous section. The only change |
+ | We again will be using Haskell arrays as the run-time representation for our vectors. In fact, the implementation of vectors is the same as that in the previous section. The only change is the use of decimal rather than unary types to describe the sizes of our vectors. |

− | is the use of decimal rather than unary types to describe the sizes of |
||

− | our vectors. |
||

<haskell> |
<haskell> |
||

module FixedDecT (..export list elided..) where |
module FixedDecT (..export list elided..) where |
||

import Data.Array |
import Data.Array |
||

</haskell> |
</haskell> |
||

− | Since we will be using the decimal notation, we need the terms and |
+ | Since we will be using the decimal notation, we need the terms and the types for all ten digits: |

− | the types for all ten digits: |
||

<haskell> |
<haskell> |
||

data D0 = D0 |
data D0 = D0 |
||

Line 346: | Line 341: | ||

data D9 = D9 |
data D9 = D9 |
||

</haskell> |
</haskell> |
||

− | For clarity and to save space, we elide repetitive code |
+ | For clarity and to save space, we elide repetitive code fragments. The [[#CodeForPaper|full code]] is available. To manipulate the digits uniformly (e.g., to find out the corresponding integer), we put them into a class <hask>Digit</hask>. We also introduce a class for non-zero digits. The latter has no methods: we use <hask>NonZeroDigit</hask> as a constraint on allowable digits. |

− | fragments. The [[#CodeForPaper|full code]] is available. To manipulate the digits uniformly (e.g., to find out the |
||

− | corresponding integer), we put them into a class <hask>Digit</hask>. We also introduce a class for non-zero digits. The latter has no |
||

− | methods: we use <hask>NonZeroDigit</hask> as a constraint on allowable |
||

− | digits. |
||

<haskell> |
<haskell> |
||

class Digit d where -- class of digits |
class Digit d where -- class of digits |
||

Line 362: | Line 357: | ||

instance NonZeroDigit D9 |
instance NonZeroDigit D9 |
||

</haskell> |
</haskell> |
||

− | We define a class of non-negative numerals. We make all |
+ | We define a class of non-negative numerals. We make all single-digit numerals the members of that class: |

− | single-digit numerals the members of that class: |
||

<haskell> |
<haskell> |
||

class Card c where |
class Card c where |
||

Line 373: | Line 368: | ||

instance Card D9 where c2num _ = 9 |
instance Card D9 where c2num _ = 9 |
||

</haskell> |
</haskell> |
||

− | We define a two-digit number, a tuple <hask>(d1,d2)</hask> |
+ | We define a two-digit number, a tuple <hask>(d1,d2)</hask> where <hask>d1</hask> is a non-zero digit, a member of the class <hask>Card</hask>. The class <hask>NonZeroDigit</hask> makes expressing the constraint lucid. We also introduce three-digit decimal numerals <hask>(d1,d2,d3)</hask>: |

− | where <hask>d1</hask> is a non-zero digit, a member of the class <hask>Card</hask>. The class <hask>NonZeroDigit</hask> makes expressing |
||

− | the constraint lucid. We also introduce three-digit decimal |
||

− | numerals <hask>(d1,d2,d3)</hask>: |
||

<haskell> |
<haskell> |
||

instance (NonZeroDigit d1,Digit d2) => Card (d1,d2) where |
instance (NonZeroDigit d1,Digit d2) => Card (d1,d2) where |
||

Line 383: | Line 378: | ||

+ (d2num $ t33 c) |
+ (d2num $ t33 c) |
||

</haskell> |
</haskell> |
||

− | The instance constraints of the <hask>Card</hask> instances |
+ | The instance constraints of the <hask>Card</hask> instances guarantee the uniqueness of our representation of numbers: the major decimal digit of a multi-digit number is not zero. It will be a type error to attempt to form such an number: |

− | guarantee the uniqueness of our representation of numbers: the |
||

− | major decimal digit of a multi-digit number is not zero. It will be a |
||

− | type error to attempt to form such an number: |
||

<haskell> |
<haskell> |
||

*FixedDecT> vec (D0,D1) 'a' |
*FixedDecT> vec (D0,D1) 'a' |
||

Line 389: | Line 384: | ||

No instance for (NonZeroDigit D0) |
No instance for (NonZeroDigit D0) |
||

</haskell> |
</haskell> |
||

− | The auxiliary compile-time functions <hask>t12</hask>...<hask>t33</hask> are tuple selectors. We could have avoided them in GHC with |
+ | The auxiliary compile-time functions <hask>t12</hask>...<hask>t33</hask> are tuple selectors. We could have avoided them in GHC with Glasgow extensions, which supports local type variables. We feel however that keeping the code Haskell98 justifies the extra hassle: |

− | Glasgow extensions, which supports local type variables. We feel |
||

− | however that keeping the code Haskell98 justifies the extra hassle: |
||

<haskell> |
<haskell> |
||

t12::(a,b) -> a; t12 = undefined |
t12::(a,b) -> a; t12 = undefined |
||

Line 403: | Line 398: | ||

listVec' size elems = Vec $ listArray (0,(c2num size)-1) elems |
listVec' size elems = Vec $ listArray (0,(c2num size)-1) elems |
||

</haskell> |
</haskell> |
||

− | The implementations of the polymorphic functions <hask>listVec</hask>, <hask>vec</hask>, <hask>vlength_t</hask>, <hask>vlength</hask>, <hask>vat</hask>, |
+ | The implementations of the polymorphic functions <hask>listVec</hask>, <hask>vec</hask>, <hask>vlength_t</hask>, <hask>vlength</hask>, <hask>vat</hask>, <hask>velems</hask>, and <hask>vzipWith</hask> are precisely the same |

− | <hask>velems</hask>, and <hask>vzipWith</hask> are precisely the same |
+ | as those in Section [[#Encoding the number parameter in type constructors, in unary|sec:unary-type]]. We elide the code for the sake of space. We introduce a few sample vectors, using the decimal notation this time: |

− | as those in Section [[#Encoding the number parameter in type constructors, in unary|sec:unary-type]]. We elide the code for |
||

− | the sake of space. We introduce a few sample vectors, using the |
||

− | decimal notation this time: |
||

<haskell> |
<haskell> |
||

v12c = listVec (D1,D2) $ take 12 ['a'..'z'] |
v12c = listVec (D1,D2) $ take 12 ['a'..'z'] |
||

Line 410: | Line 405: | ||

v13i = listVec (D1,D3) [1..13] |
v13i = listVec (D1,D3) [1..13] |
||

</haskell> |
</haskell> |
||

− | The decimal notation is so much convenient. We can now define long |
+ | The decimal notation is so much convenient. We can now define long vectors without pain. As before, the type of our vectors — the size part of the type — looks precisely the same as the corresponding size term expression: |

− | vectors without pain. As before, the type of our vectors — the size |
||

− | part of the type — looks precisely the same as the corresponding |
||

− | size term expression: |
||

<haskell> |
<haskell> |
||

*FixedDecT> :type v12c |
*FixedDecT> :type v12c |
||

Vec (D1, D2) Char |
Vec (D1, D2) Char |
||

</haskell> |
</haskell> |
||

− | We can use the [[#CodeForPaper|sample vectors]] in the tests like those of the |
+ | We can use the [[#CodeForPaper|sample vectors]] in the tests like those of the previous section. If we attempt to elementwise add two vectors of different sizes, we get a type error: |

− | previous section. If we attempt to |
||

− | elementwise add two vectors of different sizes, we get a type |
||

− | error: |
||

<haskell> |
<haskell> |
||

test5 = vzipWith (+) v12i v13i |
test5 = vzipWith (+) v12i v13i |
||

Line 425: | Line 420: | ||

In the definition of `test5': vzipWith (+) v12i v13i |
In the definition of `test5': vzipWith (+) v12i v13i |
||

</haskell> |
</haskell> |
||

− | The error message literally says that 12 is not equal to 13: the |
+ | The error message literally says that 12 is not equal to 13: the typechecker expected a vector of size 12 but found a vector of size 13 instead. |

− | typechecker expected a vector of size 12 but found a vector of size 13 |
||

− | instead. |
||

− | |||

== Arbitrary-precision decimal types == |
== Arbitrary-precision decimal types == |
||

− | From the practical point of view, the fixed-precision |
+ | |

− | number-parameterized vectors of the previous section are |
+ | From the practical point of view, the fixed-precision number-parameterized vectors of the previous section are sufficient. The imposition of a limit on the width of the decimal numerals — however easily extended — is nevertheless intellectually unsatisfying. One may wish for an encoding of arbitrarily large decimal numbers within a framework that has been set up once and for all. Such |

− | sufficient. The imposition of a limit on the width of the decimal |
+ | an SML framework has been introduced in [[#Blume01|Blume01]], to encode the sizes of arrays in their types. It is interesting to ask if such an encoding is possible in Haskell. The present section demonstrates a representation of arbitrary large decimal numbers in ''Haskell98''. We also show that typeclasses in Haskell have made the encoding easier and precise: our decimal types are in |

− | numerals — however easily extended — is nevertheless intellectually |
+ | bijection with non-negative integers. As before, we use the decimal types as phantom types describing the shape of number-parameterized vectors. |

− | unsatisfying. One may wish for an encoding of arbitrarily large decimal |
||

− | numbers within a framework that has been set up once and for all. Such |
||

− | an SML framework has been introduced in [[#Blume01|Blume01]], to |
||

− | encode the sizes of arrays in their types. It is interesting to ask |
||

− | if such an encoding is possible in Haskell. The present section |
||

− | demonstrates a representation of arbitrary large decimal numbers in |
||

− | ''Haskell98''. We also show that typeclasses in Haskell have |
||

− | made the encoding easier and precise: our decimal types are in |
||

− | bijection with non-negative integers. As before, we use the decimal |
||

− | types as phantom types describing the shape of number-parameterized |
||

− | vectors. |
||

We start by defining the types for the ten digits: |
We start by defining the types for the ten digits: |
||

Line 444: | Line 439: | ||

data D9 a = D9 a |
data D9 a = D9 a |
||

</haskell> |
</haskell> |
||

− | Unlike the code in the previous section, <hask>D0</hask> through <hask>D9</hask> are type constructors of one argument. We |
+ | Unlike the code in the previous section, <hask>D0</hask> through <hask>D9</hask> are type constructors of one argument. We use the composition of the constructors to represent sequences of digits. And so we introduce a class for arbitrary sequences of digits: |

− | use the composition of the constructors to represent sequences of |
||

− | digits. And so we introduce a class for arbitrary sequences of |
||

− | digits: |
||

<haskell> |
<haskell> |
||

class Digits ds where |
class Digits ds where |
||

ds2num:: (Num a) => ds -> a -> a |
ds2num:: (Num a) => ds -> a -> a |
||

</haskell> |
</haskell> |
||

− | with a method to convert a sequence to the corresponding |
+ | with a method to convert a sequence to the corresponding number. The method <hask>ds2num</hask> is designed in the accumulator-passing style: its second argument is the accumulator. We also need a type, which we call <hask>Sz</hask>, to represent an empty sequence of digits: |

− | number. The method <hask>ds2num</hask> is designed in the |
||

− | accumulator-passing style: its second argument is the accumulator. We |
||

− | also need a type, which we call <hask>Sz</hask>, to represent an empty |
||

− | sequence of digits: |
||

<haskell> |
<haskell> |
||

data Sz = Sz -- zero size (or the Nil of the sequence) |
data Sz = Sz -- zero size (or the Nil of the sequence) |
||

Line 467: | Line 462: | ||

t22::(f x) -> x; t22 = undefined |
t22::(f x) -> x; t22 = undefined |
||

</haskell> |
</haskell> |
||

− | The type and the term <hask>Sz</hask> is an empty sequence; |
+ | The type and the term <hask>Sz</hask> is an empty sequence; <hask>D9 Sz</hask> — that is, the application of the constructor <hask>D9</hask> to <hask>Sz</hask> — is a sequence of one digit, digit 9. The application of the constructor <hask>D1</hask> to the latter sequence gives us <hask>D1 (D9 Sz)</hask>, a two-digit sequence of digits one and nine. Compositions of data/type constructors indeed encode sequences of digits. As before, the terms and the types look precisely the same. The compositions can of course be arbitrarily long: |

− | <hask>D9 Sz</hask> — that is, the application of the constructor <hask>D9</hask> to <hask>Sz</hask> — is a sequence of one digit, digit 9. The |
||

− | application of the constructor <hask>D1</hask> to the latter sequence |
||

− | gives us <hask>D1 (D9 Sz)</hask>, a two-digit sequence of digits one |
||

− | and nine. Compositions of data/type constructors indeed encode |
||

− | sequences of digits. As before, the terms and the types look precisely |
||

− | the same. The compositions can of course be arbitrarily long: |
||

<haskell> |
<haskell> |
||

*ArbPrecDecT> :type D1$ D2$ D3$ D4$ D5$ D6$ D7$ D8$ D9$ D0$ D9$ |
*ArbPrecDecT> :type D1$ D2$ D3$ D4$ D5$ D6$ D7$ D8$ D9$ D0$ D9$ |
||

Line 477: | Line 472: | ||

1234567890987654321 |
1234567890987654321 |
||

</haskell> |
</haskell> |
||

− | We should point out a notable advantage of Haskell typeclasses in |
+ | We should point out a notable advantage of Haskell typeclasses in designing of sophisticated type families — in particular, in specifying constraints. Nothing prevents a programmer from using our type constructors, e.g., <hask>D1</hask>, in unintended ways. For example, a programmer may form a value of the type <hask>D1 Bool</hask>: either by applying a data constructor <hask>D1</hask> to a boolean value, or by casting a polymorphic value, <hask>undefined</hask>, |

− | designing of sophisticated type families — in particular, in |
||

− | specifying constraints. Nothing prevents a programmer from using our |
||

− | type constructors, e.g., <hask>D1</hask>, in unintended ways. For |
||

− | example, a programmer may form a value of the type <hask>D1 Bool</hask>: either by applying a data constructor <hask>D1</hask> to a boolean |
||

− | value, or by casting a polymorphic value, <hask>undefined</hask>, |
||

into that type: |
into that type: |
||

<haskell> |
<haskell> |
||

Line 485: | Line 480: | ||

D1 Bool |
D1 Bool |
||

</haskell> |
</haskell> |
||

− | However, such types do ''not'' represent decimal |
+ | However, such types do ''not'' represent decimal sequences. Indeed, an attempt to pass either of these values to <hask>ds2num</hask> will result in a type error: |

− | sequences. Indeed, an attempt to pass either of these values to |
||

− | <hask>ds2num</hask> will result in a type error: |
||

<haskell> |
<haskell> |
||

*ArbPrecDecT> ds2num (undefined::D1 Bool) 0 |
*ArbPrecDecT> ds2num (undefined::D1 Bool) 0 |
||

Line 492: | Line 487: | ||

In the definition of `it': ds2num (undefined :: D1 Bool) 0 |
In the definition of `it': ds2num (undefined :: D1 Bool) 0 |
||

</haskell> |
</haskell> |
||

− | In contrast, the approach in [[#Blume01|Blume01]] prevented the |
+ | In contrast, the approach in [[#Blume01|Blume01]] prevented the user from constructing (non-bottom) values of these types by a careful design and export of value constructors. That approach relied on SML’s module system to preclude the overt mis-use of the decimal type system. Yet the user can still form a (latent, in SML) bottom value of the “bad” type, e.g., by attaching an appropriate type signature to an empty list, error function or other suitable polymorphic value. In a non-strict language like Haskell such values would make our approach, which relies on phantom types, unsound. Fortunately, we are able to eliminate ill-formed decimal types at the type level rather than at the term level. Our class <hask>Digits</hask> admits those and ''only'' those types that represent sequences of digits. |

− | user from constructing (non-bottom) values of these types by a careful |
||

− | design and export of value constructors. That approach relied on SML’s |
||

− | module system to preclude the overt mis-use of the decimal type |
||

− | system. Yet the user can still form a (latent, in SML) bottom value of |
||

− | the “bad” type, e.g., by attaching an appropriate type signature to |
||

− | an empty list, error function or other suitable polymorphic value. In |
||

− | a non-strict language like Haskell such values would make our approach, |
||

− | which relies on phantom types, unsound. Fortunately, we are able to |
||

− | eliminate ill-formed decimal types at the type level rather than at |
||

− | the term level. Our class <hask>Digits</hask> admits those and ''only'' those types that represent sequences of digits. |
||

− | To guarantee the bijection between non-negative numbers and |
+ | To guarantee the bijection between non-negative numbers and sequences of digits, we need to impose an additional restriction: the first, i.e., the major, digit of a sequence must be non-zero. Expressing such a restriction is surprisingly straightforward in Haskell, even Haskell98. |

− | sequences of digits, we need to impose an additional restriction: the |
||

− | first, i.e., the major, digit of a sequence must be |
||

− | non-zero. Expressing such a restriction is surprisingly |
||

− | straightforward in Haskell, even Haskell98. |
||

<haskell> |
<haskell> |
||

class (Digits c) => Card c where |
class (Digits c) => Card c where |
||

Line 506: | Line 501: | ||

instance (Digits ds) => Card (D9 ds) |
instance (Digits ds) => Card (D9 ds) |
||

</haskell> |
</haskell> |
||

− | As in the previous sections, the class <hask>Card</hask> |
+ | As in the previous sections, the class <hask>Card</hask> represents non-negative integers. A non-negative integer is realized here as a sequence of decimal digits — provided, as the instances specify, that the sequence starts with a digit other than zero. We can now define the type of our number-parameterized vectors: |

− | represents non-negative integers. A non-negative integer is realized |
||

− | here as a sequence of decimal digits — provided, as the instances |
||

− | specify, that the sequence starts with a digit other than zero. We can |
||

− | now define the type of our number-parameterized vectors: |
||

<haskell> |
<haskell> |
||

newtype Vec size a = Vec (Array Int a) deriving Show |
newtype Vec size a = Vec (Array Int a) deriving Show |
||

Line 516: | Line 511: | ||

v13i = listVec (D1 $ D3 Sz) [1..13] |
v13i = listVec (D1 $ D3 Sz) [1..13] |
||

</haskell> |
</haskell> |
||

− | we should note a slight change of notation compared to the |
+ | we should note a slight change of notation compared to the corresponding vectors of Section [[#Fixed-precision decimal types|sec:decimal-fixed]]. The |

− | corresponding vectors of Section [[#Fixed-precision decimal types|sec:decimal-fixed]]. The |
||

tests are not changed and continue to work as before: |
tests are not changed and continue to work as before: |
||

<haskell> |
<haskell> |
||

Line 526: | Line 521: | ||

Vec (array (0,11) [(0,2),(1,4),(2,6),...(11,24)]) |
Vec (array (0,11) [(0,2),(1,4),(2,6),...(11,24)]) |
||

</haskell> |
</haskell> |
||

− | The compiler has been able to infer the size of the result of the |
+ | The compiler has been able to infer the size of the result of the <hask>vzipWith</hask> operation. The size is lucidly spelled in decimal in the type of the vector. Again, an attempt to elementwise add vectors of different sizes leads to a type error: |

− | <hask>vzipWith</hask> operation. The size is lucidly spelled in |
||

− | decimal in the type of the vector. Again, an attempt to elementwise |
||

− | add vectors of different sizes leads to a type error: |
||

<haskell> |
<haskell> |
||

test5 = vzipWith (+) v12i v13i |
test5 = vzipWith (+) v12i v13i |
||

Line 535: | Line 530: | ||

In the definition of `test5': vzipWith (+) v12i v13i |
In the definition of `test5': vzipWith (+) v12i v13i |
||

</haskell> |
</haskell> |
||

− | The typechecker complains that 2 is not equal to 3: it found the |
+ | The typechecker complains that 2 is not equal to 3: it found the vector of size 13 whereas it expected a vector of size 12. The decimal types make the error message very clear. |

− | vector of size 13 whereas it expected a vector of size 12. The decimal |
||

− | types make the error message very clear. |
||

− | We must again point out a significant difference of our approach |
+ | We must again point out a significant difference of our approach from that of [[#Blume01|Blume01]]. We were able to state that only those types of digital sequences that start with a non-zero digit correspond to a non-negative number. SML, as acknowledged in [[#Blume01|Blume01]], is unable to express such a restriction directly. The [[#Blume01|paper]], therefore, prevents the user from building invalid decimal sequences by relying on the module system: by exporting carefully-designed value constructors. The latter use an auxiliary phantom type to keep track of “nonzeroness” of the major digit. Our approach does not incur such a complication. Furthermore, by the very inductive construction of the classes <hask>Digits</hask> and <hask>Card</hask>, there is a one-to-one correspondence between ''types'', the members of <hask>Card</hask>, and the integers in decimal notation. In [[#Blume01|Blume01]], the similar mapping holds only when the family of decimal types is restricted to the types that correspond to constructible values. A user of that system may |

− | from that of [[#Blume01|Blume01]]. We were able to state that only |
+ | still form bottom values of invalid decimal types, which will cause run-time errors. In our case, when the digit constructors are misapplied, the result will no longer be in the class <hask>Card</hask>, and so the error will be detected ''statically'' by the typechecker: |

− | those types of digital sequences that start with a non-zero digit |
||

− | correspond to a non-negative number. SML, as acknowledged in [[#Blume01|Blume01]], is unable to express such a restriction directly. The |
||

− | [[#Blume01|paper]], therefore, prevents the user from building |
||

− | invalid decimal sequences by relying on the module system: by |
||

− | exporting carefully-designed value constructors. The latter use an |
||

− | auxiliary phantom type to keep track of “nonzeroness” of the major |
||

− | digit. Our approach does not incur such a complication. Furthermore, |
||

− | by the very inductive construction of the classes <hask>Digits</hask> |
||

− | and <hask>Card</hask>, there is a one-to-one correspondence between |
||

− | ''types'', the members of <hask>Card</hask>, and the integers |
||

− | in decimal notation. In [[#Blume01|Blume01]], the similar mapping |
||

− | holds only when the family of decimal types is restricted to the types |
||

− | that correspond to constructible values. A user of that system may |
||

− | still form bottom values of invalid decimal types, which will cause |
||

− | run-time errors. In our case, when the digit constructors are |
||

− | misapplied, the result will no longer be in the class <hask>Card</hask>, and so the error will be detected ''statically'' by the |
||

− | typechecker: |
||

<haskell> |
<haskell> |
||

*ArbPrecDecT> vec (D1$ D0$ D0$ True) 0 |
*ArbPrecDecT> vec (D1$ D0$ D0$ True) 0 |
||

Line 553: | Line 548: | ||

== Computations with decimal types == |
== Computations with decimal types == |
||

+ | |||

The previous sections gave many examples of functions such as |
The previous sections gave many examples of functions such as |
||

<hask>vzipWith</hask> that take two vectors ''statically'' |
<hask>vzipWith</hask> that take two vectors ''statically'' |
||

Line 966: | Line 962: | ||

using the versions of <hask>vhead</hask> and <hask>vtail</hask> |
using the versions of <hask>vhead</hask> and <hask>vtail</hask> |
||

without any further run-time size checks. |
without any further run-time size checks. |
||

− | |||

== Related work == |
== Related work == |
||

+ | |||

This paper was inspired by Matthias Blume’s messages on the |
This paper was inspired by Matthias Blume’s messages on the |
||

newsgroup comp.lang.functional in February 2002. Many ideas |
newsgroup comp.lang.functional in February 2002. Many ideas |
||

Line 1,118: | Line 1,114: | ||

functions’ types. Another approach to multivariate functions — |
functions’ types. Another approach to multivariate functions — |
||

multivariate composition operator — is discussed in [[#mcomp|mcomp]]. |
multivariate composition operator — is discussed in [[#mcomp|mcomp]]. |
||

− | |||

== Conclusions == |
== Conclusions == |
||

+ | |||

Throughout this paper we have demonstrated several realizations of |
Throughout this paper we have demonstrated several realizations of |
||

number-parameterized types in Haskell, using arrays parameterized by |
number-parameterized types in Haskell, using arrays parameterized by |
||

Line 1,149: | Line 1,145: | ||

addition and subtraction of sizes — and presents |
addition and subtraction of sizes — and presents |
||

readable error messages on violation of the constraints. |
readable error messages on violation of the constraints. |
||

− | |||

## Revision as of 20:57, 8 August 2012

**This article needs reformatting! Please help tidy it up.**--WouterSwierstra 14:26, 9 May 2008 (UTC)

# 1 Number-parameterized types

*This article is also available in PDF. This Wiki page is not the master file: rather, it is the result of the SXML->Wiki conversion.*

# 2 Abstract

This paper describes practical programming with types
parameterized by numbers: e.g., an array type parameterized by the
array’s size or a modular group type `Zn`
parameterized by the modulus. An attempt to add, for example, two
integers of different moduli should result in a compile-time error
with a clear error message. Number-parameterized types let the
programmer capture more invariants through types and eliminate some
run-time checks.

We review several encodings of the numeric
parameter but concentrate on the phantom type representation of a
sequence of *decimal* digits. The decimal encoding makes
programming with number-parameterized types convenient and error
messages more comprehensible. We implement arithmetic on
decimal number-parameterized types, which lets us statically
typecheck operations such as array concatenation.

Overall we demonstrate a practical
dependent-type-like system that is just a Haskell library. The basics
of the number-parameterized types are written in Haskell98.

### 2.1 Keywords:

Haskell, number-parameterized types, type arithmetic, decimal types, type-directed programming.

# 3 Contents

## 3.1 Introduction

Discussions about types parameterized by values — especially types of arrays or finite groups parameterized by their size — reoccur every couple of months on functional programming languages newsgroups and mailing lists. The often expressed wish is to guarantee that, for example, we never attempt to add two vectors of different lengths. As one poster said, “This [feature] would be helpful in the crypto library where I end up having to either define new length Words all the time or using lists and losing the capability of ensuring I am manipulating lists of the same length.” Number-parameterized types as other more expressive types let us tell the typechecker our intentions. The typechecker may then help us write the code correctly. Many errors (which are often trivial) can be detected at compile time. Furthermore, we no longer need to litter the code with array boundary match checks. The code therefore becomes more readable, reliable, and fast. Number-parameterized types when expressed in signatures also provide a better documentation of the code and let the invariants be checked across module boundaries.

In this paper, we develop realizations of number-parameterized types in Haskell that indeed have all the above advantages. The numeric parameter is specified in *decimal* rather than in binary, which makes types smaller and far easier to read. Type error messages also become more comprehensible. The programmer may write or the compiler can infer equality constraints (e.g., two argument

Although we come close to the dependent-type programming, we do not extend either a compiler or the language. Our system is a regular Haskell library. In fact, the basic number-parameterized types can be implemented entirely in Haskell98. Advanced operations such as type arithmetic require commonly supported Haskell98 extensions to multi-parameter classes with functional dependencies and higher-ranked types.

Our running example is arrays parameterized over their size. The parameter of the vector type is therefore a non-negative integer number. For simplicity, all the vectors in the paper are indexed from zero. In addition to vector constructors and element accessors, we define aThe present paper describes several gradually more sophisticated number-parameterized Haskell libraries. We start by paraphrasing the approach by Chris Okasaki, who represents the size parameter of vectors in a sequence of data constructors. We then switch to the encoding of the size in a sequence of type constructors. The resulting types are phantom and impose no run-time overhead. Section sec:unary-type describes unary encoding of numerals in type constructors, Sections sec:decimal-fixed and sec:decimal-arb discuss decimal encodings. Section sec:decimal-fixed introduces a type representation for fixed-precision decimal numbers. Section sec:decimal-arb removes the limitation on the maximal size of representable numbers, at a cost of a more complex implementation and of replacing commas with unsightly dollars signs. The decimal encoding is extendible to other bases, e.g., 16 or 64. The latter can be used to develop practical realizations of number-parameterized cryptographically interesting groups.

Section sec:arithmetic describes the first contribution of the paper. We develop addition and subtraction of “decimal types”, i.e., of the type constructor applications representing non-negative integers in decimal notation. The implementation is significantly different from that for more common unary numerals. Although decimal numerals are notably difficult to add, they make number-parameterized programming practical. We can now write arithmetic equality and inequality constraints on number-parameterized types.

Section sec:dynamic briefly describes working with number-parameterized types when the numeric parameter, and even its upper bound, are not known until run time. We show one, quite simple technique, which assures a static constraint by a run-time check — witnessing. The witnessing code, which must be trustworthy, is notably compact. The section uses the method of blending of static and dynamic assurances that was first described in stanamic-trees.

Section sec:related compares our approach with the phantom type programming in SML by Matthias Blume, with a practical dependent-type system of Hongwei Xi, with statically-sized and generic arrays in Pascal and C, with the shape inference in array-oriented languages, and with C++ template meta-programming. Section sec:conclusions concludes.

## 3.2 Encoding the number parameter in data constructors

The first approach to vectors parameterized by their size encodes the size as a series of data constructors. This approach has been used extensively by Chris Okasaki. For example, in Okasaki99 he describes square matrixes whose dimensions can be proved equal at compile time. He digresses briefly to demonstrate vectors of statically known size. A similar technique has been described by McBride. In this section, we develop a more naive encoding of the size through data constructors, for introduction and comparison with the encoding of the size via type constructors in the following sections.

Our representation of vectors of a statically checked size is reminiscent of the familiar representation of lists:

data List a = Nil | Cons a (List a)

module UnaryDS where data VZero a = VZero deriving Show infixr 3 :+: data Vecp tail a = a :+: (tail a) deriving Show

*type*.

To generically manipulate the family of differently-sized vectors, we define a class of polymorphic functions:

class Vec t where vlength:: t a -> Int vat:: t a -> Int -> a vzipWith:: (a->b->c) -> t a -> t b -> t c

instance Vec VZero where vlength = const 0 vat = error "null array or index out of range" vzipWith f a b = VZero instance (Vec tail) => Vec (Vecp tail) where vlength (_ :+: t) = 1 + vlength t vat (a :+: _) 0 = a vat (_ :+: ta) n = vat ta (n-1) vzipWith f (a :+: ta) (b :+: tb) = (f a b) :+: (vzipWith f ta tb)

That was the complete implementation of the number-parameterized vectors. We can now define a few sample vectors:

v3c = 'a' :+: 'b' :+: 'c' :+: VZero v3i = 1 :+: 2 :+: 3 :+: VZero v4i = 1 :+: 2 :+: 3 :+: 4 :+: VZero

and a few simple tests:

test1 = vlength v3c test2 = [vat v3c 0, vat v3c 1, vat v3c 2]

We can load the code into a Haskell system and run the tests. Incidentally, we can ask the Haskell system to tell us the inferred type of a sample vector:

*UnaryDS> :t v3c Vecp (Vecp (Vecp VZero)) Char

The inferred type indeed encodes the size of the vector as a Peano numeral. We can try more complex tests, of element-wise operations on two vectors:

test3 = vzipWith (\c i -> (toEnum $ fromEnum c + fromIntegral i)::Char) v3c v3i test4 = vzipWith (+) v3i v3i *UnaryDS> test3 'b' :+: ('d' :+: ('f' :+: VZero))

An attempt to add, by mistake, two vectors of different sizes is revealing:

test5 = vzipWith (+) v3i v4i Couldn't match `VZero' against `Vecp VZero' Expected type: Vecp (Vecp (Vecp VZero)) a Inferred type: Vecp (Vecp (Vecp (Vecp VZero))) a1 In the third argument of `vzipWith', namely `v4i' In the definition of `test5': vzipWith (+) v3i v4i

We get a type error, with a clear error message (the quoted message, here and elsewhere in the paper, is by GHCi. The Hugs error message is essentially the same). The typechecker, at the compile time, has detected that the sizes of the vectors to add elementwise do not match. To be more precise, the sizes are off by one.

For vectors described in this section, the element access operation,`O(n)`time where

`n`is the size of the vector. Chris Okasaki has designed more sophisticated number-parameterized vectors with element access time

`O(log n)`. Although this is an improvement, the overhead of accessing an element adds up for many operations. Furthermore, the overhead of data constructors,

## 3.3 Encoding the number parameter in type constructors, in unary

To improve the efficiency of number-parameterized vectors, we choose a better run-time representation: Haskell arrays. The code in the present section is in Haskell98.

module UnaryT (..elided..) where import Data.Array

First, we need a type structure (an infinite family of types) to encode non-negative numbers. In the present section, we will use an unary encoding in the form of Peano numerals. The unary type encoding of integers belongs to programming folklore. It is also described in Blume01 in the context of a foreign-function interface library of SML.

data Zero = Zero data Succ a = Succ a

`n`is represented as a repeated application of

`n`type (data) constructors

class Card c where c2num:: (Num a) => c -> a -- convert to a number cpred::(Succ c) -> c cpred = undefined instance Card Zero where c2num _ = 0 instance (Card c) => Card (Succ c) where c2num x = 1 + c2num (cpred x)

The same correspondence between the types and the terms suggests that the numeral type alone is enough to describe the size of a vector. We do not need to store the value of the numeral. The shape type of our vectors could be *phantom* (as in Blume01).

newtype Vec size a = Vec (Array Int a) deriving Show

listVec':: (Card size) => size -> [a] -> Vec size a listVec' size elems = Vec $ listArray (0,(c2num size)-1) elems listVec:: (Card size) => size -> [a] -> Vec size a listVec size elems | not (c2num size == length elems) = error "listVec: static/dynamic sizes mismatch" listVec size elems = listVec' size elems vec:: (Card size) => size -> a -> Vec size a vec size elem = listVec' size $ repeat elem

*UnaryT> listVec (Succ (Succ Zero)) [True,False] Vec (array (0,1) [(0,True),(1,False)])

A Haskell interpreter created the requested value, and printed it out. We can confirm that the inferred type of the vector encodes its size:

*UnaryT> :type listVec (Succ (Succ Zero)) [True,False] Vec (Succ (Succ Zero)) Bool

We can now introduce functions to operate on our vectors. The functions are similar to those in the previous section. As before, they are polymorphic in the shape of vectors (i.e., their sizes). This polymorphism is expressed differently however. In the present section we use just the parametric polymorphism rather than typeclasses.

vlength_t:: Vec size a -> size vlength_t _ = undefined vlength:: Vec size a -> Int vlength (Vec arr) = let (0,last) = bounds arr in last+1 velems:: Vec size a -> [a] velems (Vec v) = elems v vat:: Vec size a -> Int -> a vat (Vec arr) i = arr ! i vzipWith:: Card size => (a->b->c) -> Vec size a -> Vec size b -> Vec size c vzipWith f va vb = listVec' (vlength_t va) $ zipWith f (velems va) (velems vb)

*compile-time*function. Therefore, it is no surprise that its body is

*is*its true definition. The function

infixl 3 &+ data VC size a = VC size [a] vs:: VC Zero a; vs = VC Zero [] (&+):: VC size a -> a -> VC (Succ size) a (&+) (VC size lst) x = VC (Succ size) (x:lst) vc:: (Card size) => VC size a -> Vec size a vc (VC size lst) = listVec' size (reverse lst)

v3c = vc $ vs &+ 'a' &+ 'b' &+ 'c' v3i = vc $ vs &+ 1 &+ 2 &+ 3 v4i = vc $ vs &+ 1 &+ 2 &+ 3 &+ 4 test1 = vlength v3c; test1' = vlength_t v3c test2 = [vat v3c 0, vat v3c 1, vat v3c 2] test3 = vzipWith (\c i -> (toEnum $ fromEnum c + fromIntegral i)::Char) v3c v3i test4 = vzipWith (+) v3i v3i

We can run the tests as follows:

*UnaryT> test3 Vec (array (0,2) [(0,'b'),(1,'d'),(2,'f')]) *UnaryT> :type test3 Vec (Succ (Succ (Succ Zero))) Char

The type of the result bears the clear indication of the size of the vector. If we attempt to perform an element-wise operation on vectors of different sizes, for example:

test5 = vzipWith (+) v3i v4i Couldn't match `Zero' against `Succ Zero' Expected type: Vec (Succ (Succ (Succ Zero))) a Inferred type: Vec (Succ (Succ (Succ (Succ Zero)))) a1 In the third argument of `vzipWith', namely `v4i' In the definition of `test5': vzipWith (+) v3i v4i

we get a message from the typechecker that the sizes are off by one.

## 3.4 Fixed-precision decimal types

Peano numerals adequately represent the size of a vector in vector’s type. However, they make the notation quite verbose. We want to offer a programmer a familiar, decimal notation for the terms and the types representing non-negative numerals. This turns out possible even in Haskell98. In this section, we describe a fixed-precision notation, assuming that a programmer will never need a vector with more than 999 elements. The limit is not hard and can be readily extended. The next section will eliminate the limit altogether.

We again will be using Haskell arrays as the run-time representation for our vectors. In fact, the implementation of vectors is the same as that in the previous section. The only change is the use of decimal rather than unary types to describe the sizes of our vectors.

module FixedDecT (..export list elided..) where import Data.Array

Since we will be using the decimal notation, we need the terms and the types for all ten digits:

data D0 = D0 data D1 = D1 ... data D9 = D9

class Digit d where -- class of digits d2num:: (Num a) => d -> a -- convert to a number instance Digit D0 where d2num _ = 0 instance Digit D1 where d2num _ = 1 ... instance Digit D9 where d2num _ = 9 class Digit d => NonZeroDigit d instance NonZeroDigit D1 instance NonZeroDigit D2 ... instance NonZeroDigit D9

We define a class of non-negative numerals. We make all single-digit numerals the members of that class:

class Card c where c2num:: (Num a) => c -> a -- convert to a number -- Single-digit numbers are non-negative numbers instance Card D0 where c2num _ = 0 instance Card D1 where c2num _ = 1 ... instance Card D9 where c2num _ = 9

instance (NonZeroDigit d1,Digit d2) => Card (d1,d2) where c2num c = 10*(d2num $ t12 c) + (d2num $ t22 c) instance (NonZeroDigit d1,Digit d2,Digit d3) => Card (d1,d2,d3) where c2num c = 100*(d2num $ t13 c) + 10*(d2num $ t23 c) + (d2num $ t33 c)

*FixedDecT> vec (D0,D1) 'a' <interactive>:1: No instance for (NonZeroDigit D0)

t12::(a,b) -> a; t12 = undefined t22::(a,b) -> b; t22 = undefined ... t33::(a,b,c) -> c; t33 = undefined

The rest of the code is as before, e.g.:

newtype Vec size a = Vec (Array Int a) deriving Show listVec':: Card size => size -> [a] -> Vec size a listVec' size elems = Vec $ listArray (0,(c2num size)-1) elems

as those in Section sec:unary-type. We elide the code for the sake of space. We introduce a few sample vectors, using the decimal notation this time:

v12c = listVec (D1,D2) $ take 12 ['a'..'z'] v12i = listVec (D1,D2) [1..12] v13i = listVec (D1,D3) [1..13]

The decimal notation is so much convenient. We can now define long vectors without pain. As before, the type of our vectors — the size part of the type — looks precisely the same as the corresponding size term expression:

*FixedDecT> :type v12c Vec (D1, D2) Char

We can use the sample vectors in the tests like those of the previous section. If we attempt to elementwise add two vectors of different sizes, we get a type error:

test5 = vzipWith (+) v12i v13i Couldn't match `D2' against `D3' Expected type: Vec (D1, D2) a Inferred type: Vec (D1, D3) a1 In the third argument of `vzipWith', namely `v13i' In the definition of `test5': vzipWith (+) v12i v13i

The error message literally says that 12 is not equal to 13: the typechecker expected a vector of size 12 but found a vector of size 13 instead.

## 3.5 Arbitrary-precision decimal types

From the practical point of view, the fixed-precision number-parameterized vectors of the previous section are sufficient. The imposition of a limit on the width of the decimal numerals — however easily extended — is nevertheless intellectually unsatisfying. One may wish for an encoding of arbitrarily large decimal numbers within a framework that has been set up once and for all. Such
an SML framework has been introduced in Blume01, to encode the sizes of arrays in their types. It is interesting to ask if such an encoding is possible in Haskell. The present section demonstrates a representation of arbitrary large decimal numbers in *Haskell98*. We also show that typeclasses in Haskell have made the encoding easier and precise: our decimal types are in
bijection with non-negative integers. As before, we use the decimal types as phantom types describing the shape of number-parameterized vectors.

We start by defining the types for the ten digits:

module ArbPrecDecT (..export list elided..) where import Data.Array data D0 a = D0 a data D1 a = D1 a ... data D9 a = D9 a

class Digits ds where ds2num:: (Num a) => ds -> a -> a

data Sz = Sz -- zero size (or the Nil of the sequence) instance Digits Sz where ds2num _ acc = acc

We now inductively define arbitrarily long sequences of digits:

instance (Digits ds) => Digits (D0 ds) where ds2num dds acc = ds2num (t22 dds) (10*acc) instance (Digits ds) => Digits (D1 ds) where ds2num dds acc = ds2num (t22 dds) (10*acc + 1) ... instance (Digits ds) => Digits (D9 ds) where ds2num dds acc = ds2num (t22 dds) (10*acc + 9) t22::(f x) -> x; t22 = undefined

*ArbPrecDecT> :type D1$ D2$ D3$ D4$ D5$ D6$ D7$ D8$ D9$ D0$ D9$ D8$ D7$ D6$ D5$ D4$ D3$ D2$ D1$ Sz D1 (D2 (D3 (D4 (D5 (D6 (D7 (D8 (D9 (D0 (D9 (D8 (D7 (D6 (D5 (D4 (D3 (D2 (D1 Sz)))))))))))))))))) *ArbPrecDecT> ds2num (D1$ D2$ D3$ D4$ D5$ D6$ D7$ D8$ D9$ D0$ D9$ D8$ D7$ D6$ D5$ D4$ D3$ D2$ D1$ Sz) 0 1234567890987654321

into that type:

*ArbPrecDecT> :type D1 True D1 Bool *ArbPrecDecT> :type (undefined::D1 Bool) D1 Bool

*not*represent decimal sequences. Indeed, an attempt to pass either of these values to

*ArbPrecDecT> ds2num (undefined::D1 Bool) 0 No instance for (Digits Bool) arising from use of `ds2num' at <interactive>:1 In the definition of `it': ds2num (undefined :: D1 Bool) 0

*only*those types that represent sequences of digits.

To guarantee the bijection between non-negative numbers and sequences of digits, we need to impose an additional restriction: the first, i.e., the major, digit of a sequence must be non-zero. Expressing such a restriction is surprisingly straightforward in Haskell, even Haskell98.

class (Digits c) => Card c where c2num:: (Num a) => c -> a c2num c = ds2num c 0 instance Card Sz instance (Digits ds) => Card (D1 ds) instance (Digits ds) => Card (D2 ds) ... instance (Digits ds) => Card (D9 ds)

newtype Vec size a = Vec (Array Int a) deriving Show

v12c = listVec (D1 $ D2 Sz) $ take 12 ['a'..'z'] v12i = listVec (D1 $ D2 Sz) [1..12] v13i = listVec (D1 $ D3 Sz) [1..13]

we should note a slight change of notation compared to the corresponding vectors of Section sec:decimal-fixed. The tests are not changed and continue to work as before:

test4 = vzipWith (+) v12i v12i *ArbPrecDecT> :type test4 Vec (D1 (D2 Sz)) Int *ArbPrecDecT> test4 Vec (array (0,11) [(0,2),(1,4),(2,6),...(11,24)])

test5 = vzipWith (+) v12i v13i Couldn't match `D2 Sz' against `D3 Sz' Expected type: Vec (D1 (D2 Sz)) a Inferred type: Vec (D1 (D3 Sz)) a1 In the third argument of `vzipWith', namely `v13i' In the definition of `test5': vzipWith (+) v12i v13i

The typechecker complains that 2 is not equal to 3: it found the vector of size 13 whereas it expected a vector of size 12. The decimal types make the error message very clear.

We must again point out a significant difference of our approach from that of Blume01. We were able to state that only those types of digital sequences that start with a non-zero digit correspond to a non-negative number. SML, as acknowledged in Blume01, is unable to express such a restriction directly. The paper, therefore, prevents the user from building invalid decimal sequences by relying on the module system: by exporting carefully-designed value constructors. The latter use an auxiliary phantom type to keep track of “nonzeroness” of the major digit. Our approach does not incur such a complication. Furthermore, by the very inductive construction of the classes*types*, the members of

*statically*by the typechecker:

*ArbPrecDecT> vec (D1$ D0$ D0$ True) 0 No instance for (Digits Bool) arising from use of `vec' at <interactive>:1 In the definition of `it': vec (D1 $ (D0 $ (D0 $ True))) 0 *ArbPrecDecT> vec (D0$ D1$ D0 Sz) 0 No instance for (Card (D0 (D1 (D0 Sz)))) arising from use of `vec' at <interactive>:1 In the definition of `it': vec (D0 $ (D1 $ (D0 Sz))) 0

## 3.6 Computations with decimal types

The previous sections gave many examples of functions such as

*statically*

known to be of equal size. The signature of these functions states
quite detailed invariants whose violations will be reported at
compile-time. Furthermore, the invariants can be inferred by the
compiler itself. This use of the type system is not particular to
Haskell: Matthias Blume has derived a similar
parameterization of arrays in SML, which can express such equality of
size constraints. Matthias Blume however cautions one not to overstate
the usefulness of the approach because the type system can express
only fairly simple constraints: “There is still no type that, for
example, would force two otherwise arbitrary arrays to differ in size
by exactly one.” That was written in the context of SML however. In
Haskell with common extensions we *can* define vector
functions whose type contains arithmetic constraints on the sizes of
the argument and the result vectors. These constraints can be verified
statically and sometimes even inferred by a compiler. In this section,
we consider the example of vector concatenation. We shall see that the

of the result is the sum of the sizes of two argument vectors. We also

introduce the functionswhose type specifies that they can only be applied to non-empty

vectors. Furthermore, the type ofsize of the result vector is less by one than the size of the argument vector. These examples are quite unusual and almost cross into the realm of dependent types.

We must note however that the examples in this section require the Haskell98 extension to multi-parameter classes with functional

dependencies. That extension is activated by flagsGHCi.

We will be using the arbitrary precision decimal types introduced in the previous section. We aim to design a ‘type addition’ of decimal sequences. Our decimal types spell the corresponding non-negative numbers in the conventional (i.e., big-endian) decimal notation: the most-significant digit first. However, it is more convenient to add such numbers starting from the least-significant digit. Therefore, we need a way to reverse digital sequences, or more precise, types of the

classalgorithm written in the accumulator-passing style.

class DigitsInReverse' df w dr | df w -> dr instance DigitsInReverse' Sz acc acc instance (Digits (d drest), DigitsInReverse' drest (d acc) dr) => DigitsInReverse' (d drest) acc dr

instances spell this constraint out. We can now introduce a class that relates a sequence of digits with its reverse:

class DigitsInReverse df dr | df -> dr, dr -> df instance (DigitsInReverse' df Sz dr, DigitsInReverse' dr Sz df) => DigitsInReverse df dr

each other. The functional dependencies make the “each other” part clear: one sequence uniquely determines the other. The typechecker

will verify that givendigits_rev:: (Digits ds, Digits dsr, DigitsInReverse ds dsr) => ds -> dsr digits_rev = undefined

It is again a compile-time function specified entirely by its type. Its body is therefore undefined. We can now run a few examples:

*ArbArithmT> :t digits_rev (D1$D2$D3 Sz) D3 (D2 (D1 Sz)) *ArbArithmT> :t (\v -> digits_rev v `asTypeOf` (D1$D2$D3 Sz)) D3 (D2 (D1 Sz)) -> D1 (D2 (D3 Sz))

Indeed, the process of reversing sequences of decimal digits works

both ways. Given the type of the argument tothe compiler infers the type of the result. Conversely, given the type of the result the compiler infers the type of the argument.

A sequence of digits belongs to the classif the most-significant digit is not a zero. To convert an arbitrary

sequence toclass NoLeadingZeros d d0 | d -> d0 instance NoLeadingZeros Sz Sz instance (NoLeadingZeros d d') => NoLeadingZeros (D0 d) d' instance NoLeadingZeros (D1 d) (D1 d) ... instance NoLeadingZeros (D9 d) (D9 d)

We are now ready to build the addition machinery. We draw our inspiration from the computer architecture: the adder of an arithmetical-logical unit (ALU) of the CPU is constructed by chaining of so-called full-adders. A full-adder takes two summands and the carry-in and yields the result of the summation and the carry-out. In our case, the summands and the result are decimal rather than binary. Carry is still binary.

class FullAdder d1 d2 cin dr cout | d1 d2 cin -> cout, d1 d2 cin -> dr, d1 dr cin -> cout, d1 dr cin -> d2 where _unused:: (d1 xd1) -> (d2 xd2) -> cin -> (dr xdr) _unused = undefined

constructors. The functional dependencies of the class tell us that the summands and the input carry uniquely determine the result digit and the output carry. On the other hand, if we know the result digit,

one of the summands,can therefore be used for addition and for subtraction. In the latter case, the carry bits should be more properly called borrow bits.

data Carry0 data Carry1 instance FullAdder D0 D0 Carry0 D0 Carry0 instance FullAdder D0 D0 Carry1 D1 Carry0 instance FullAdder D0 D1 Carry0 D1 Carry0 ... instance FullAdder D9 D8 Carry1 D8 Carry1 instance FullAdder D9 D9 Carry0 D8 Carry1 instance FullAdder D9 D9 Carry1 D9 Carry1

The full code indeed contains 200 instances of

functional dependencies of the class. The number of instances could be significantly reduced if we availed ourselves to an overlapping instances extension. For generality however we tried to use as few Haskell98 extensions as possible. Although 200 instances seems like quite many, we have to write them only once. We place the instances into a module and separately compile it. Furthermore, we did not write those instances by hand: we used Haskell itself:

make_full_adder = mapM_ putStrLn [unwords $ doit d1 d2 cin | d1<-[0..9], d2<-[0..9], cin<-[0..1]] where doit d1 d2 cin = ["instance FullAdder", tod d1, tod d2, toc cin, tod d12, toc cout] where (d12,cout) = let sum = d1 + d2 + cin in if sum >= 10 then (sum-10,1) else (sum,0) tod n | (n >= 0 && 9 >= n) = "D" ++ (show n) toc 0 = "Carry0"; toc 1 = "Carry1"

That function is ready for Template Haskell. Currently we used a low-tech approach of cutting and pasting from an Emacs buffer with GHCi into the Emacs buffer with the code.

We useonwards, and we propagate the carry. If one input sequence turns out shorter than the other, we pad it with zeros. The correctness of the algorithm follows by simple induction.

class DigitsSum ds1 ds2 cin dsr | ds1 ds2 cin -> dsr instance DigitsSum Sz Sz Carry0 Sz instance DigitsSum Sz Sz Carry1 (D1 Sz) instance (DigitsSum (D0 Sz) (d2 d2rest) cin (d12 d12rest)) => DigitsSum Sz (d2 d2rest) cin (d12 d12rest) instance (DigitsSum (d1 d1rest) (D0 Sz) cin (d12 d12rest)) => DigitsSum (d1 d1rest) Sz cin (d12 d12rest) instance (FullAdder d1 d2 cin d12 cout, DigitsSum d1rest d2rest cout d12rest) => DigitsSum (d1 d1rest) (d2 d2rest) cin (d12 d12rest)

subtraction to be non-negative.

class DigitsDif ds1 ds2 cin dsr | ds1 dsr cin -> ds2 instance DigitsDif Sz ds Carry0 ds instance (DigitsDif (D0 Sz) (d2 d2rest) Carry1 (d12 d12rest)) => DigitsDif Sz (d2 d2rest) Carry1 (d12 d12rest) instance (FullAdder d1 d2 cin d12 cout, DigitsDif d1rest d2rest cout d12rest) => DigitsDif (d1 d1rest) (d2 d2rest) cin (d12 d12rest)

together:

class (Card c1, Card c2, Card c12) => CardSum c1 c2 c12 | c1 c2 -> c12, c1 c12 -> c2 instance (Card c1, Card c2, Card c12, DigitsInReverse c1 c1r, DigitsInReverse c2 c2r, DigitsSum c1r c2r Carry0 c12r, DigitsDif c1r c2r' Carry0 c12r, DigitsInReverse c2r' c2', NoLeadingZeros c2' c2, DigitsInReverse c12r c12) => CardSum c1 c2 c12

that the latter is the sum of the formers. The two summands determine the sum, or the sum and one summand determine the other. The class can be used for addition and subtraction of sequences. The dependencies of

the solereverse the summand sequences to make them little-endian, add them together with the zero carry, and reverse the result. We also make sure that the subtraction and summation are the exact inverses. The

addition algorithmwith the major digit zero. The subtraction algorithm however may result in a sequence with zero major digits, which have to be stripped

away, with the help of the relationthe addition out:

card_sum:: CardSum c1 c2 c12 => c1 -> c2 -> c12 card_sum = undefined

*ArbArithmT> :t card_sum (D1 Sz) (D9$D9 Sz) D1 (D0 (D0 Sz)) *ArbArithmT> :t \v -> card_sum (D1 Sz) v `asTypeOf` (D1$D0$D0 Sz) D9 (D9 Sz) -> D1 (D0 (D0 Sz)) *ArbArithmT> :t \v -> card_sum (D9$D9 Sz) v `asTypeOf` (D1$D0$D0 Sz) D1 Sz -> D1 (D0 (D0 Sz))

The typechecker can indeed add and subtract with carry and

borrow. Now we define the functionconcatenate two vectors.

vappend va vb = listVec (card_sum (vlength_t va) (vlength_t vb)) $ (velems va) ++ (velems vb)

we chose however to perform a run-time check and avoid proving the theorem about the size of the list concatenation result. We did not declare

the type of*ArbArithmT> :t vappend vappend :: (CardSum size size1 c12) => Vec size a -> Vec size1 a -> Vec c12 a

which literally says that the size of the result vector is the sum of the sizes of the argument vectors. The constraint is spelled out

patently, as part of the type ofbe arbitrarily large decimal numbers: for example, the following expression demonstrates the concatenation of a vector of 25 elements and a vector of size 979:

*ArbArithmT> :t vappend (vec (D2$D5 Sz) 0) (vec (D9$D7$D9 Sz) 0) (Num a) => Vec (D1 (D0 (D0 (D4 Sz)))) a

Blume01 as an unattainable wish.

vhead:: CardSum (D1 Sz) size1 size => Vec size a -> Vec (D1 Sz) a vhead va = listVec (D1 Sz) $ [head (velems va)] vtail:: CardSum (D1 Sz) size1 size => Vec size a -> Vec size1 a vtail va = result where result = listVec (vlength_t result) $ tail (velems va)

function result, the function is not divergent and not

recursive. Recall thatessentially specifies an *inequality* constraint: the input
vector is non-empty. The constraint is expressed via an implicitly

size of the input vector.

We can now run a few examples. We note that the compiler could correctly infer the type of the result, which includes the size of the vector after appending or truncating it.

*ArbArithmT> let v = vappend (vec (D9 Sz) 0) (vec (D1 Sz) 1) *ArbArithmT> :t v Vec (D1 (D0 Sz)) Integer *ArbArithmT> v Vec (array (0,9) [(0,0),(1,0),...,(8,0),(9,1)]) *ArbArithmT> :type vhead v Vec (D1 Sz) Integer *ArbArithmT> :type vtail v Vec (D9 Sz) Integer *ArbArithmT> vtail v Vec (array (0,8) [(0,0),(1,0),...,(7,0),(8,1)]) *ArbArithmT> :type (vappend (vhead v) (vtail v)) Vec (D1 (D0 Sz)) Integer

non-empty argument vector constraint. Indeed, an attempt to apply

*ArbArithmT> vtail (vec Sz 0) <interactive>:1:0: No instances for (DigitsInReverse' c2' Sz c2r', DigitsInReverse' c2r' Sz c2', DigitsDif (D1 Sz) c2r' Carry0 Sz, DigitsSum (D1 Sz) c2r Carry0 Sz, DigitsInReverse' c2r Sz size1, DigitsInReverse' size1 Sz c2r) arising from use of `vtail' at <interactive>:1:0-4

The error message essentially says that there is no such decimal

typeholds. That is, there is no non-negative number that gives zero if added to one.

We can form quite complex expressions from the functionscompiler will *infer* and verify the corresponding
constraints on the sizes of involved vectors. For example:

testc1 = let va = vec (D1$D2 Sz) 0 vb = vec (D5 Sz) 1 vc = vec (D8 Sz) 2 in vzipWith (+) va (vappend vb (vtail vc)) *ArbArithmT> testc1 Vec (array (0,11) [(0,1),...,(4,1),(5,2),(6,2),...,(11,2)])

inferred this non-trivial constraint and checked it. Indeed, if we by

mistake writewriting the example, the compiler will instantly report a type error:

Couldn't match `D9 Sz' against `D8 Sz' Expected type: D9 Sz Inferred type: D8 Sz When using functional dependencies to combine DigitsSum (D1 Sz) c2r Carry0 (D9 Sz), arising from use of `vtail' at ArbArithmT.hs:420:34-38 DigitsSum (D1 Sz) c2r Carry0 (D8 Sz), arising from use of `vtail' at ArbArithmT.hs:411:34-38

We can define other operations that extend or shrink our vectors. For example, Section sec:unary-type introduced

the operatoreasier. It is straightforward to implement such an operator for decimally-typed vectors.

We must point out that the type system guarantees thatvectors. Therefore, we no longer need the corresponding run-time

check. The bodies of*safely*use unsafe versions of the library functions

of the code without compromising its safety.

## 3.7 Statically-sized vectors in a dynamic context

In the present version of the paper, we demonstrate the simplest method of handling number-parameterized vectors in the dynamic context. The method involves run-time checks. The successful result of a run-time check is marked with the appropriate static type. Further computations can therefore rely on the result of the check (e.g., that the vector in question definitely has a particular size) and avoid the need to do that test over and over again. The net advantage is the reduction in the number of run-time checks. The complete elimination of the run-time checks is quite difficult (in general, may not even be possible) and ultimately requires a dependent type system.

For our presentation we use an example of dynamically-sized vectors: reversing a vector by the familiar accumulator-passing algorithm. Each iteration splits the source vector into the head and the tail, and prepends the head to the accumulator. The sizes of the vectors change in the course of the computation, to be precise, on each iteration. We treat vectors as if they were lists. Most of the vector processing code does not have such a degree of variation in vector sizes. The code is quite simple:

vreverse v = listVec (vlength_t v) $ reverse $ velems v

whose inferred type is obviously

*ArbArithmT> :t vreverse vreverse :: (Card size) => Vec size a -> Vec size a

of the input vector. We do this test only once, at the conclusion of the algorithm. We can treat the result as any other number-parameterized vector, for example:

testv = let v = vappend (vec (D3 Sz) 1) (vec (D1 Sz) 4) vr = vreverse v in vhead (vtail (vtail vr))

without any further run-time size checks.

## 3.8 Related work

This paper was inspired by Matthias Blume’s messages on the
newsgroup comp.lang.functional in February 2002. Many ideas
of this paper were first developed during the USENET discussion, and
posted in a series of three messages at that time. In more detail
Matthias Blume described his method in Blume01,
although that paper uses binary rather than decimal types of array
sizes for clarity. The approaches by Matthias Blume and ours both rely on
phantom types to encode additional information about a value (e.g.,
the size of an array) in a manner suitable for a typechecker. The
paper exhibits the most pervasive and thorough
use of phantom types: to represent the size of arrays and the
constness of imported C values, to encode C structure tag *names* and C function prototypes.

However, paper was written in the context of SML, whereas we use Haskell. The language has greatly influenced the method of specifying and enforcing complex static constraints, e.g., that digit sequences representing non-negative numbers must not have leading zeros. The SML approach in Blume01 relies on the sophisticated module system of SML to restrict the availability of value constructors so that users cannot build values of outlawed types. Haskell typeclasses on the other hand can directly express the constraint, as we saw in Section sec:decimal-arb. Furthermore, Haskell typeclasses let us specify arithmetic equality and inequality constraints — which, as admitted in Blume01, seems quite unlikely to be possible in SML.

Arrays of a statically known size — whose size is a part of their type — are a fairly popular feature in programming languages. Such arrays are present in Fortran, Pascal, C <ref>C does permit truly statically-sized arrays like those in Pascal. To achieve this, we should make a C array a member of a C structure. The compiler preserves the array size information when passing such a wrapped array as an argument. It is even possible to assign such “arrays”.</ref>. Pascal has the most complete realization of statically sized arrays. A Pascal compiler can therefore typecheck array functions like

ourexpressiveness and efficiency: for example, in Pascal we can copy one instance of an array into another instance of the same type by a single assignment, which, for small arrays, can be fully inlined by the compiler into a sequential code with no loops or range checks. However, in a language without the parametric polymorphism statically sized arrays are a great nuisance. If the size of an array is a part of its type, we cannot write generic functions that operate on arrays of any size. We can only write functions dealing with arrays of specific, fixed sizes. The inability to build generic array-processing libraries is one of the most serious drawbacks of Pascal. Therefore, Fortran and C introduce “generic” arrays whose size type is not statically known. The compiler silently converts a statically-sized array into a generic one when passing arrays as arguments to functions. We can now build generic array-processing libraries. We still need to know the size of the array. In Fortran and C, the programmer must arrange for passing the size information to a function in some other way, e.g., via an additional argument, global variable, etc. It becomes then the responsibility of a programmer to make sure that the size information is correct. The large number of Internet security advisories related to buffer overflows and other array-management issues testify that programmers in general are not to be relied upon for correctly passing and using the array size information. Furthermore, the silent, irreversible conversion of statically sized arrays into generic ones negate all the benefits of the former.

A different approach to array processing is a so-called shape-invariant programming, which is a key feature of array-oriented languages such as APL or SaC. These languages let a programmer define operations that can be applied to arrays of arbitrary shape/dimensionality. The code becomes shorter and free from explicit iterations, and thus more reusable, easier to read and to write. The exact shape of an array has to be known, eventually. Determining it at run-time is greatly inefficient. Therefore, high-performance array-oriented languages employ shape inference Scholz01, which tries to statically infer the dimensionalities or even exact sizes of all arrays in a program. Shape inference is, in general, undecidable, since arrays may be dynamically allocated. Therefore, one can either restrict the class of acceptable shape-invariant programs to a decidable subset, resort to a dependent-type language like Cayenne, or use “soft typing”. The latter approach is described in Scholz01, which introduces a non-unique type system based on a hierarchy of array types: from fully specialized ones with the statically known sizes and dimensionality, to a type of an array with the known dimensionality but not size, to a fully generic array type whose shape can only be determined at run-time. The system remains decidable because at any time the typechecker can throw up hands and give to a value a fully generic array type. Shape inference of SaC is specific to that language, whose type system is otherwise deliberately constrained: SaC lacks parametric polymorphism and higher-order functions. Using shape inference for compilation of shape-invariant array operations into a highly efficient code is presented in Kreye. Their compiler tries to generate as precise shape-specific code as possible. When the shape inference fails to give the exact sizes or dimensionalities, the compiler emits code for a dynamic shape dispatch and generic loops.

There is however a great difference in goals and implementation between the shape inference of SaC and our approach. The former aims at accepting more programs than can statically be inferred shape-correct. We strive to express assertions about the array sizes and enforcing the programming style that assures them. We have shown

the definitions of functions such asargument and the result vectors are all of the same size. This constraint is assured at compile-time — even if we do not statically know the exact sizes of the vectors. Because SaC lacks parametric polymorphism, it cannot express such an assertion and statically

verify it. If a SaC programmer applies a function such asthat as an error but will compile a generic array code instead. The error will be raised at run time during a range check.

The approach of the present paper comes close to emulating a dependent type system, of which Cayenne is the epitome. We were particularly influenced by a practical dependent type system of Hongwei Xi Xi98 XiThesis, which is a conservative extension of SML. In Xi98, Hongwei Xi et al. demonstrated an application of their system to the elimination of array bound checking and list tag checking. The related work section of that paper lists a number of other dependent and pseudo-dependent type systems. Using the type system to avoid unnecessary run-time checks is a goal of the present paper too.

C++ templates provide parametric polymorphism and indexing of types by true integers. A C++ programmer can therefore define

functions likeequality and even arithmetic constraints on the sizes of the argument vectors. Blitz++ was the first example of using a so-called template meta-programming for generating efficient and safe array code. The type system of C++ however presents innumerable hurdles to the functional style. For example, the result type of a function is not used for the overloading resolution, which significantly restricts the power of the type inference. Templates were introduced in C++ ad hoc, and therefore, are not well integrated with its type system. Violations of static constraints expressed via templates result in error messages so voluminous as to become incomprehensible.

McBride gives an extensive survey of the emulation of dependent type systems in Haskell. He also describes number-parameterized arrays that are similar to the ones discussed in Section sec:Okasaki. The paper by Fridlender and Indrika shows another example of emulating dependent types within the Hindley-Milner type system: namely, emulating

variable-arity functions such as generictechnique relies on ad hoc codings for natural numbers which resemble Peano numerals. They aim at defining more functions (i.e., multi-variate functions), whereas we are concerned with making functions more restrictive by expressing sophisticated invariants in functions’ types. Another approach to multivariate functions — multivariate composition operator — is discussed in mcomp.

## 3.9 Conclusions

Throughout this paper we have demonstrated several realizations of number-parameterized types in Haskell, using arrays parameterized by their size as an example. We have concentrated on techniques that rely on phantom types to encode the size information in the type of the array value. We have built a family of infinite types so that different values of the vector size can have their own distinct type. That type is a decimal encoding of the corresponding integer (rather than the more common unary, Peano-like encoding). The examples throughout the paper illustrate that the decimal notation for the number-parameterized vectors makes our approach practical.

We have used the phantom size types to express non-trivial constraints on the sizes of the argument and the result arrays in the type of functions. The constraints include the size equality, e.g., the type of a function of two arguments may indicate that the arguments must be vectors of the same size. More importantly, we can specify arithmetical constraints: e.g., that the size of the vector after concatenation is the sum of the source vector sizes. Furthermore, we can write inequality constraints by means of an implicit existential quantification, e.g., the function

programmer should benefit from more expressive function signatures and from the ability of the compiler to statically check complex invariants in all applications of the vector-processing functions. The compiler indeed infers and checks non-trivial constraints involving addition and subtraction of sizes — and presents readable error messages on violation of the constraints.

# 4 References

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Matthias Blume: No-Longer-Foreign: Teaching an ML compiler to speak C “natively.” In BABEL’01: First workshop on multi-language infrastructure and interoperability, September 2001, Firenze, Italy. [1]

The complete source code for the article. August 9, 2005. [2]

Daniel Fridlender and Mia Indrika: Do we Need Dependent Types? BRICS Report Series RS-01-10, March 2001. [3]

Oleg Kiselyov: Polyvariadic composition. October 31, 2003. [4]

Oleg Kiselyov: Polymorphic stanamically balanced AVL trees. April 26, 2003. [5]

Dietmar Kreye: A Compilation Scheme for a Hierarchy of Array Types. Proc. 3th International Workshop on Implementation of Functional Languages (IFL’01).

Conor McBride: Faking it — simulating dependent types in Haskell. Journal of Functional Programming, 2002, v.12, pp. 375-392 [6]

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